Strong unicity criterion in some space of operators

Let X be a finite dimensional Banach space and let Y ⊂ X be a hyperplane. Let L Y = {L ∈ L (X, Y ) : L |Y = 0}. In this note, we present sufficient and necessary conditions on L0 ∈ L Y being a strongly unique best approximation for given L ∈ L (X). Next we apply this characterization to the case of X = l ∞ and to generalization of Theorem I.1.3 from [12] (see also [13]).